# Full text of "Treatise On Analysis Vol-Ii"

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```1    DEFINITION OF A MEASURE      99

and the properties of measures defined by densities (13.15) give us a method
for dealing with denumerable families of measures on a locally compact
space, by fixing a basic measure and working with the densities relative to this
basic measure (hence again with functions); this again proves to be extremely
convenient. Here the modern point of view emerges: given a /^-integrable
function/, what is important is not the values taken by/so much as the way
in which /operates on the space of bounded continuous functions by means
of the linear mapping #h-» \fg d\i (this mapping depends only on the equiva-
lence class of/and therefore does not change when we modify/at the points
of a set of measure zero). The development of this point of view will lead in
Chapter XVII to the theory of distributions, which is a natural generalization
of the notion of measure on differential manifolds.

Throughout this chapter, the phrase "locally compact space" will always
mean "separable metrizable locally compact space."

1. DEFINITION  OF A MEASURE

To begin with, let X be a compact (metrizable) space. A measure (or
complex measure) on X is by definition an element of the dual of the Banach
space ^cW of complex-valued continuous functions on X (7.2), that is to
say ((12.15) and (5.5.1)) it is a linear form /h-> /*(/) on ^C(X) which satisfies an
inequality of the form

(13.1.1)

for all/e ^C(X) (recall that ||/|| = sup

xeX

Now let X be a locally compact space (metrizable and separable, in
accordance with our conventions). For every compact subset K of X, let
JT(X; K) (or Jf*c(X; K)) denote the vector subspace of ^C(X) consisting of
the functions whose support (12.6) is contained in K (and is therefore compact).
We shall denote by Jfc(X) (or Jf(X)) the union of the Jfc(X; K) as K runs
through all compact subsets of X. In other words, JfcW is the vector space
of (complex-valued) continuous functions with compact support. Clearly
JTC(X) c «£(X).

A measure (or complex measure) on X is by definition a linear form \JL on
«?TC(X) with the following property: for each compact subset K of X, there
exists a real number aK ^ 0 (in general depending on K) such that

(13.12)
forall/ejf(X;K).                      J r                                 /
```